NAME DATE PERIOD. Study Guide and Intervention

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1 5- Study Guide and Intervention Long Division To divide a polynomial by a monomial, use the skills learned in Lesson 5-1. To divide a polynomial by a polynomial, use a long division pattern. Remember that only like terms can be added or subtracted. Example 1 1 p t r - 1 p qt r - 9 p tr p tr Simplify 1 p t r - 1 p qt r - 9 p tr p tr = 1 p t r p tr - 1 p qt r p tr. - 9p tr p tr = 1 p ( - ) t ( - 1) r (1-1) - 1 p ( - ) qt (1-1) r ( - 1) - 9 p ( - ) t (1-1) (1-1) r = 4pt -7qr - p Example x - 4x - 1 x - 4 ) x - 8x + 4x - 9 (-) x - 4x -4x + 4x (-)-4x + 16x -1x - 9 (-)-1x + 48 Use long division to find (x - 8x + 4x - 9) (x - 4). -57 The quotient is x - 4x - 1, and the remainder is -57. Therefore x - 8 x + 4x - 9 x - 4 Exercises a + 0 a a = x - 4x x m n 6-40 m n 4 m n 4. (x - 5x - ) (x - ) 5. (m - m - 7) (m + ) 6. (p - 6) (p - 1) 7. (t - 6t + 1) (t + ). 60 a b - 48 b a 5 b 1a b Lesson X (x 5-1) (x - 1) 9. (x - 5x + 4x - 4) (x - ) Chapter 5 11 Glencoe Algebra

2 5- Study Guide and Intervention Synthetic Division (continued) Synthetic division a procedure to divide a polynomial by a binomial using coefficients of the dividend and the value of r in the divisor x - r Use synthetic division to find (x - 5x + 5x - ) (x - 1). Step 1 Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients. x - 5x + 5x Step Write the constant r of the divisor x - r to the left, In this case, r = 1. Bring down the fi rst coefficient,, as shown Step Step 4 Step 5 Multiply the first coefficient by r, 1 =. Write their product under the second coefficient. Then add the product and the second coefficient: -5 + = -. Multiply the sum, -, by r: - 1 = -. Write the product under the next coefficient and add: 5 + (-) =. Multiply the sum,, by r: 1 =. Write the product under the next coefficient and add: - + = 0. The remainder is Thus, (x - 5x + 5x - ) (x - 1) = x -. Exercises 1. (x - 7x + 9x - 14) (x - ). (5x + 7x - x - ) (x + 1). (x + x - 10x - ) (x + ) 4. (x - 8x + 19x - 9) (x - 4) 5. (x + 10x + 9x + 8) (x + 5) 6. (x - 8x + 16x - 1) (x - 1) 7. (x - 9x + 17x - 1) (x - ) 8. (4x - 5x + 40) (x - 6) 9. (6x + 8x - 7x + 9) (x + 5) 10. (x 4-4x + x + 7x - ) (x - ) 11. (1x 4 + 0x - 4x + 0x + 5) (x + 5) Chapter 5 1 Glencoe Algebra

3 5- Skills Practice 1. 10c y + 6 y + y y 4. 1 x - 4x - 8 4x 5. (15q 6 + 5q )(5q 4 ) (4f 5-6f 4 + 1f - 8f )(4f ) (6j k - 9jk ) jk 8. (4a h - 8a h + a 4 ) (a ) 9. (n + 7n + 10) (n + 5) 10. (d + 4d + ) (d + 1) 11. (t + 1t + 15) (t + 5) 1. (6y + y - )(y - 1) -1 Lesson X (4g - 9) (g + ) 14. (x - 5x - 4) (x - ) 15. u + 5u - 1 u x + 5x + 9 x (v - 7v - 10)(v - 4) (t 4 + 4t - t - 5t - 0)(t + 4) y - y - 6 y + 0. x - x - 19x + 15 x - 1. (4p - p + p) (p - 1). (c 4 + 6c - c + 4)(c + ) -1. GEOMETRY The area of a rectangle is x + 8x + 1x - 1 square units. The width of the rectangle is x + 4 units. What is the length of the rectangle? Chapter 5 1 Glencoe Algebra

4 5- Practice r 10-5 r r 5 r 4. 6 k m - 1 k m + 9 m k m. (-0x y + 1x y - 18x y) (-6x y) 4. (-6w z 4 - w z 5 + 4w + 5z) (w z) 5. (4a - 8a + a )(4a) (8d k + d k - 4dk )(4dk ) f + 7f + 10 f + 8. x + x - 14 x - 9. (a - 64) (a - 4) 10. (b + 7) (b + ) 11. x + 6x + 15 x x + 4x - 6 x + 1. (w + 7w - 4w + ) (w + ) 14. (6y y - 8y - 6) (y + ) 15. (x 4 - x - 11x + x + 10) (x - 5) 16. (m 5 + m - 1) (m + 1) 17. (x 4 - x + 5x - 6)() (6y - 5y - 15)(y + ) x - x + 6 x x - x - 7 x (r + 5r - r - 15) (r - ). (6t + 5t - t + 1) (t + 1). 4 p 4-17 p + 14p - p - 4. h 4 - h + h + h - h GEOMETRY The area of a rectangle is x - 11x + 15 square feet. The length of the rectangle is x - 5 feet. What is the width of the rectangle? 6. GEOMETRY The area of a triangle is 15x 4 + x + 4x - x - square meters. The length of the base of the triangle is 6x - meters. What is the height of the triangle? Chapter 5 14 Glencoe Algebra

5 5- Word Problem Practice 1. REMAINDERS Jordan divided the polynomial x 4 + x - 6 into the polynomial p(x) yesterday. Today his work is smudged and he cannot read p(x) or most of his answer. The only part he could read was the remainder x + 4. His teacher wants him to find p(-). What is p(-)? 4. VOLUME The volume of one column of the Lincoln Memorial is π(x - x - 4x + 640). If the height of the column is x + 40 feet, find the area of the base of the column in terms of x and π.. LONG DIVISION Dana used long division to divide x 4 + x + x + x + 1 by. Her work is shown below with three numbers missing. x - x + x - 5 ) x 4 + x + x + x + 1 (-)x 4 + x -x + A (-)-x + x x + x (-)x + B -5x + 1 (-)-5x - 10 C What are A, B, and C?. AVERAGES Shelby is a statistician. She has a list of n + 1 numbers and she needs to find their average. Two of the numbers are n and. Each of the other n - 1 numbers are all equal to 1. What is the average of these numbers? 5. NUMBER THEORY Mr. Collins has his class working with bases and polynomials. He wrote on the board that the number 1111 in base B has the value B + B + B + 1. The class was then given the following questions to answer. a. The number 11 in base B has the value B + 1. What is 1111 (in base B) divided by 11 (in base B)? b. The number 111 in base B has the value B + B + 1. What is 1111 (in base B) divided by 111 (in base B)? Lesson X- 5- Chapter 5 15 Glencoe Algebra

6 5- Enrichment Oblique Asymptotes The graph of y = ax + b, where a 0, is called an oblique asymptote of y = f(x) if the graph of f comes closer and closer to the line as x or x -. is the mathematical symbol for infinity, which means endless. For f(x) = x x, y = x + 4 is an oblique asymptote because f(x) - x - 4 = x, and x 0 as x or -. In other words, as x increases, the value of x gets smaller and smaller approaching 0. Example Find the oblique asymptote for f(x) = x + 8x Use synthetic division y = x - 8x + 15 = x As x increases, the value of gets smaller. In other words, since 0 as x or x -, y = x + 6 is an oblique asymptote. Exercises Use synthetic division to find the oblique asymptote for each function. 1. y = 8 x - 4x + 11 x + 5. y = x + x - 15 x -. y = x - x - 18 x - 4. y = a x + bx + c x - d 5. y = a x + bx + c x + d Chapter 5 16 Glencoe Algebra